Trying to move Time Machine backup to a new machine

Okay, I've gotten sufficiently confused trying to pick out the process in this thread.

I need to find a step-by-step and whether I start fresh with my Mac. I've tried using the migration from my old TM drive and as I said I have 2 backup folders, after trying it twice following "How to move your content to a new Mac."

After quite a bit of frustration I'm leaning towards just doing a clean install and letting all the old stuff go. Not ideal and not utilizing what I thought TM was for, but I want to move on. I think you understand what I'm trying to do by this point. If you have any further ideas, I'm happy to try. Otherwise, clean install.

Thanks for all your help so far.

Thanks I did this, but maybe too late.

Here's the full situation: Bought new Mac and had to get to work before I had time to migrate from old Mac (which Time Machine has backed up on an external drive.) After a couple weeks worth of work on the new Mac, had time to do the migration. Old back up HD was starting to make noises like it was going to fail, so I purchased a new one. Copied Backups.backupb to new HD, in Time Machine on new Mac, selected new HD as my backup disk. Ran Time Machine and got a second folder in Backups.backupb (GregRov's iMac and GregRov's iMac 2). Able to browse other backup disk to get to old folders but can't restore from there. Is it possible to have one continuous backup? Do I need to start from scratch on my new Mac? I am buying another external HD for work so I could copy everything off, wipe things clean and do the migration at the beginning. I'd like to have this set up right instead of working around my rushed and sloppy set up.

I do have my old backup–on the original backup drive and then new one I copied it to. As I said, I worked a job on my new Mac before I did the migration to set it up. Are you saying to erase my new Mac? (If so, I can store new data from the job to a third drive temporarily, then copy it back after set up.)

I'm hoping to have Time Machine backup from my old Mac to be accessible from my new Mac, if that's possible. My old external backup drive may be failing so I want to move to a new external drive. Can Time Machine pick up on my old Mac's backup and continue with it on the new Mac so it's one seamless backup??

Since I've had the new Mac for only a few weeks, there's not that much on it. It wouldn't be much work to save the new files to a drive and then start from scratch. That way I don't need to compromise. Does this make sense?

I can easily backtrack as I have the original external backup drive, the new external drive (where I copied my TM backup, cuz the old one is making noises), and the old Mac still works. I have all the original pieces. I never backed up the new Mac to the old drive. Only the new drive has the two folders in the backup. I think it will be worth the time to set it up. The worse that can happen is I get to this same point and manually move things like you suggest. I bought the new machine for a job and didn't have time to do the set up the right way. Now I have plenty of time.

Thanks for your help!